Effect of reflector curvature on a plane wave

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.4

Redraw Figure 6.4a for a plane wave incident on the reflector, and explain the significance of the changes which this makes.

Background

Figure 6.4a assumes a point source at the surface, whereas for an incident plane wave in Figure 6.4b the source is at infinity. The plane wave reflected by the plane reflector produces a plane wavefront (R), i.e., the reflected wavefront has zero curvature. For a point diffractor the virtual source is at the diffractor and the wavefront has the maximum curvature (D). The curvature of the wavefront from the anticlinal reflector (A) is intermediate between those of $R$ and $D$ and the curvature of the wavefront from the synclinal reflector (S) is negative (assuming the center of curvature is below the surface).

Figure 6.4a.  Reflector curvature effects for point source.

Figure 6.4b.  Reflection curvature for incident plane wave.

Solution

By Huygens’s principle (problem 3.1), a diffracting point acts as a point source whenever a wave falls upon it; hence, the diffraction response $D$ to a plane wave (Figure 6.4b) is the same as that in Figure 6.4a. A plane wave incident on a plane reflector gives rise to a reflected plane wave $R$. For a plane wave incident on an anticline or syncline of circular cross-section of radius $r$, we can use the mirror formula, namely,

${\begin{aligned}1/f=2/r=1/u+1/v,\end{aligned}}$

where $f$ is the focal length $=r/2$, $u$ and $v$ are the distances of the source and reflected image from the apex of an anticline or trough of a syncline; $r$ is positive for a syncline and $u$ is infinite for a plane wave, so $v=r/2$. For a syncline, the reflected wave comes to a focus a distance $r/2$ above the trough. For an anticline, a virtual image (see problem 4.1) is at $r/2$ below the high point.

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Table of Contents (book)

Also in this chapter

• Characteristics of different types of events and noise
• Horizontal resolution
• Reflection and refraction laws and Fermat’s principle
• Effect of reflector curvature on a plane wave
• Diffraction traveltime curves
• Amplitude variation with offset for seafloor multiples
• Ghost amplitude and energy
• Directivity of a source plus its ghost
• Directivity of a harmonic source plus ghost
• Differential moveout between primary and multiple
• Suppressing multiples by NMO differences
• Distinguishing horizontal/vertical discontinuities
• Identification of events
• Traveltime curves for various events
• Reflections/diffractions from refractor terminations
• Refractions and refraction multiples
• Destructive and constructive interference for a wedge
• Dependence of resolvable limit on frequency
• Vertical resolution
• Causes of high-frequency losses
• Ricker wavelet relations
• Improvement of signal/noise ratio by stacking

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